Partial Differential Equations 2: Functional Analytic Methods

✍ Scribed by Sauvigny, Friedrich

• Publisher: Springer
• Year: 2012
• Tongue: English
• Leaves: 464
• Series: Universitext
• Category: Library

✦ Synopsis

In two comprehensive volumes, updated and revised in a second edition, this textbook spans elliptic, parabolic, and hyperbolic types, and several variables. This second part emphasizes functional analytic methods and applications to differential geometry.

✦ Table of Contents

Partial Differential Equations 2......Page 4
Preface – Volume 2......Page 6
Contents – Volume 2: Functional Analytic Methods and Differential Geometric Applications......Page 11
Contents – Volume 1: Foundations and Integral Representations......Page 14
1 Fixed Point Theorems......Page 16
2 The Leray-Schauder Degree of Mapping......Page 27
3 Fundamental Properties for the Degree of Mapping......Page 33
4 Linear Operators in Banach Spaces......Page 38
5 Some Historical Notices to the Chapters 3 and 7......Page 45
1 Various Eigenvalue Problems......Page 47
2 Singular Integral Equations......Page 61
3 The Abstract Hilbert Space......Page 70
4 Bounded Linear Operators in Hilbert Spaces......Page 80
5 Unitary Operators......Page 92
6 Completely Continuous Operators in Hilbert Spaces......Page 103
7 Spectral Theory for Completely Continuous Hermitian Operators......Page 120
8 The Sturm-Liouville Eigenvalue Problem......Page 127
9 Weyl's Eigenvalue Problem for the Laplace Operator......Page 134
10 Some Historical Notices to Chapter 8......Page 142
1 The Differential Equation Delta phi+p(x,y)phi x+q(x,y)phi y=r(x,y)......Page 144
2 The Schwarzian Integral Formula......Page 150
3 The Riemann-Hilbert Boundary Value Problem......Page 153
4 Potential-theoretic Estimates......Page 161
5 Schauder's Continuity Method......Page 173
6 Existence and Regularity Theorems......Page 178
7 The Schauder Estimates......Page 186
8 Some Historical Notices to Chapter 9......Page 202
1 Sobolev Spaces......Page 204
2 Embedding and Compactness......Page 218
3 Existence of Weak Solutions......Page 225
4 Boundedness of Weak Solutions......Page 229
5 Hölder Continuity of Weak Solutions......Page 232
6 Weak Potential-theoretic Estimates......Page 244
7 Boundary Behavior of Weak Solutions......Page 251
8 Equations in Divergence Form......Page 255
9 Green's Function for Elliptic Operators......Page 261
10 Spectral Theory of the Laplace-Beltrami Operator......Page 270
11 Some Historical Notices to Chapter 10......Page 273
1 The Fundamental Forms and Curvatures of a Surface......Page 274
2 Two-dimensional Parametric Integrals......Page 280
3 Quasilinear Hyperbolic Differential Equations and Systems of Second Order (Characteristic Parameters)......Page 288
4 Cauchy's Initial Value Problem for Quasilinear Hyperbolic Differential Equations and Systems of Second Order......Page 296
5 Riemann's Integration Method......Page 306
6 Bernstein's Analyticity Theorem......Page 310
7 Some Historical Notices to Chapter 11......Page 316
1 Maximum Principles for the H-surface System......Page 318
2 Gradient Estimates for Nonlinear Elliptic Systems......Page 325
3 Global Estimates for Nonlinear Systems......Page 338
4 The Dirichlet Problem for Nonlinear Elliptic Systems......Page 342
5 Distortion Estimates for Plane Elliptic Systems......Page 350
6 A Curvature Estimate for Minimal Surfaces......Page 358
7 Global Estimates for Conformal Mappings with respect to Riemannian Metrics......Page 363
8 Introduction of Conformal Parameters into a Riemannian Metric......Page 371
9 The Uniformization Method for Quasilinear Elliptic Differential Equations......Page 377
10 Some Historical Notices to Chapter 12......Page 378
1 The Dirichlet Problem for Graphs of Prescribed Mean Curvature......Page 380
2 Winding Staircases and Stable Minimal Graphs on a Riemann Surface......Page 391
3 Mixed Boundary Value Problem for Minimal Graphs over the Etale Plane......Page 401
4 Plateau's Problem with Constant Mean Curvature......Page 414
5 Closed Surfaces of Constant Mean Curvature......Page 420
6 One-to-one Solutions of Lewy-Heinz Systems......Page 429
7 On Weyl's Embedding Problem and Elliptic Monge-Ampère Equations......Page 446
8 Some Historical Notices to Chapter 13......Page 454
References......Page 457
Index......Page 460

✦ Subjects

Science;Mathematics